Pretriangulated Category Research Articles

Semiseparable Functors and Conditions up to Retracts

In a previous paper we introduced the concept of semiseparable functor. Here we continue our study of these functors in connection with idempotent (Cauchy) completion. To this aim, we introduce and investigate the notions of (co)reflection and bireflection up to retracts. We show that the (co)comparison functor attached to an adjunction whose associated (co)monad is separable is a coreflection (reflection) up to retracts. This fact allows us to prove that a right (left) adjoint functor is semiseparable if and only if the associated (co)monad is separable and the (co)comparison functor is a bireflection up to retracts, extending a characterization pursued by X.-W. Chen in the separable case. Finally, we provide a semi-analogue of a result obtained by P. Balmer in the framework of pre-triangulated categories.

Pre-(n + 2)-angulated categories

In this article, we introduce the notion of a pre-(n+2)-angulated category as a higher dimensional analogue of a pre-triangulated category defined by Beligiannis-Reiten. We first show that the idempotent completion of a pre-(n+2)-angulated category admits a unique pre-(n+2)-angulated structure. Let (C,E,s) be an n-exangulated category and X be a strongly functorially finite subcategory of C. We then show that the quotient category C/X is a pre-(n+2)-angulated category. These results allow to construct several examples of pre-(n+2)-angulated categories. Moreover, we also give a necessary and sufficient condition for the quotient C/X to be an (n+2)-angulated category.

Reflective and coreflective subcategories

Given any additive category C with split idempotents, pseudokernels and pseudocokernels, we show that a subcategory B is coreflective if, and only if, it is precovering, closed under direct summands and each morphism in B has a pseudocokernel in C that belongs to B. We apply this result and its dual to, among others, preabelian and pretriangulated categories. As a consequence, we show that a subcategory of a preabelian category is coreflective if, and only it, it is precovering and closed under taking cokernels. On the other hand, if C is pretriangulated with split idempotents, then a subcategory B is coreflective and invariant under the suspension functor if, and only if, it is precovering and closed under taking direct summands and cones. These are extensions of well-known results for AB3 abelian and triangulated categories, respectively.By-side applications of these results allow us: a) To characterize the coreflective subcategories of a given AB3 abelian category which have a set of generators and are themselves abelian, abelian exact or module categories; b) to extend to module categories over arbitrary small preadditive categories a result of Gabriel and De la Peña stating that all fully exact subcategories are bireflective; c) to show that, in any Grothendieck category, the direct limit closure of its subcategory of finitely presented objects is a coreflective subcategory.

Quotient categories of extriangulated categories

Let be a subcategory of an extriangulated category and let be a strongly functorially finite subcategory of Then the subfactor category admits a pre-triangulated structure in the sense of Beligiannis-Reiten where has the following properties: For any -triangle in where is -monic, the third term C is also in For any -triangle in where is -epic, the first term A is also in Moreover, if the above pre-triangulated category with becomes a triangulated category, then forms an -mutation pair and is closed under extensions.

Tannaka duality for enhanced triangulated categories I: reconstruction

We develop Tannaka duality theory for dg categories. To any dg functor from a dg category $\mathcal A$ to finite-dimensional complexes, we associate a dg coalgebra $C$ via a Hochschild homology construction. When the dg functor is faithful, this gives a quasi-equivalence between the derived dg categories of $\mathcal A$-modules and of $C$-comodules. When $\mathcal A$ is Morita fibrant (i.e. an idempotent-complete pre-triangulated category), it is thus quasi-equivalent to the derived dg category of compact $C$-comodules. We give several applications for motivic Galois groups.

Triangulated quotient categories revisited

Extriangulated categories were introduced by Nakaoka and Palu by extracting the similarities between exact categories and triangulated categories. A notion of mutation of subcategories in an extriangulated category is defined in this paper. Let A be an extension closed subcategory of an extriangulated category C. Then the additive quotient category M:=A/[X] carries naturally a triangulated structure whenever (A,A) forms an X-mutation pair. This result generalizes many results of the same type for triangulated categories. It is used to give a classification of thick triangulated subcategories of pre-triangulated category C/[X], where X is functorially finite in C. When C has Auslander–Reiten translation τ, we prove that for a functorially finite subcategory X of C containing projectives and injectives, the quotient C/[X] is a triangulated category if and only if (C,C) is X-mutation, and if and only if τX_=X‾. This generalizes a result by Jørgensen who proved the equivalence between the first and the third conditions for triangulated categories. Furthermore, we show that for such a subcategory X of the extriangulated category C, C admits a new extriangulated structure such that C is a Frobenius extriangulated category. Applications to exact categories and triangulated categories are given. From the applications we present extriangulated categories which are neither exact categories nor triangulated categories.

A Note on the Localization of Pretriangulated Categories

For a localizing class $$ \mathcal{S} $$ of morphisms in a pretriangulated category $$ \mathcal{D} $$ , a weak version of a sufficient condition that guarantees the transfer of the structure of a pretriangulated category onto the localization $$ \mathcal{D}\left[{\mathcal{S}}^{-1}\right] $$ is proposed. Moreover, the weakness of a similar sufficient condition is obtained in the context of triangulated categories.

Subfactor Categories of Triangulated Categories

Let 𝒳 ⊂ 𝒜 be subcategories of a triangulated category 𝒯, and 𝒳 a functorially finite subcategory of 𝒜. If 𝒜 has the properties that any 𝒳-monomorphism of 𝒜 has a cone and any 𝒳-epimorphism has a cocone, then the subfactor category 𝒜/[𝒳] forms a pretriangulated category in the sense of [4]. Moreover, the above pretriangulated category 𝒜/[𝒳] with 𝒯(𝒳, 𝒳[1]) = 0 becomes a triangulated category if and only if (𝒜, 𝒜) forms an 𝒳-mutation pair and 𝒜 is closed under extensions.

Torsion Pairs in Stable Categories

Let 𝒞 be a triangulated category. When ω is a functorially finite subcategory of 𝒞, Jøtrgensen showed that the stable category 𝒞/ω is a pretriangulated category. A pair (𝒳, 𝒴) of subcategories of 𝒞 with ω ⊆ 𝒳 ∩ 𝒴 gives rise to a pair (𝒳/ω, 𝒴/ω) of subcategories of 𝒞/ω. In this article, we find conditions for (𝒳/ω, 𝒴/ω) to be a torsion pair in terms of properties of the pair (𝒳, 𝒴). In particular, we obtain necessary and sufficient conditions for (𝒳/ω, 𝒴/ω) to be a torsion pair in the stable category 𝒞/ω when τω = ω, where τ is the Auslander–Reiten translation.

The homotopy category and derived category of N-complexes

In this paper complexes with N-nilpotent differentials are considered. We describe the loop functor Ω and the suspension functor Σ in the category CN(A) of N-complexes on an abelian category A to provide an effective construction of left and right triangles in the homotopy category KN(A) of N-complexes, and prove that KN(A) together with the functors Ω, Σ and the classes of left and right triangles mentioned above is a pretriangulated category. We also investigate different homologies and get another example of pretriangulated categories, i.e., the derived category DN(A) of N-complexes. Some basic properties are given.

Idempotent completion of pretriangulated categories

A pretriangulated category is an additive category with left and right triangulations such that these two triangulations are compatible. In this paper, we first show that the idempotent completion of a left triangulated category admits a unique structure of left triangulated category and dually this is true for a right triangulated category. We then prove that the idempotent completion of a pretriangulated category has a natural structure of pretriangulated category. As an application, we show that a torsion pair in a pretriangulated category extends uniquely to a torsion pair in the idempotent completion.

On splitting idempotents in pre-triangulated categories

Le and Chen’s theorem on splitting of idempotents is extended to the case of pre-triangulated categories equipped with a bounded t-structure

On one-sided torsion pair

Motivated by the concept of a torsion pair in a pre-triangulated category induced by Beligiannis and Reiten, the notion of a left (right) torsion pair in the left (right) triangulated category is introduced and investigated. We provide new connections between different aspects of torsion pairs in one-sided triangulated categories, pre-triangulated categories, stable categories and derived categories.

Homological and homotopical aspects of torsion theories

Introduction Torsion pairs in abelian and triangulated categories Torsion pairs in pretriangulated categories Compactly generated torsion pairs in triangulated categories Hereditary torsion pairs in triangulated categories Torsion pairs in stable categories Triangulated torsion(-free) classes in stable categories Gorenstein categories and (co)torsion pairs Torsion pairs and closed model structures (Co)torsion pairs and generalized Tate-Vogel cohomology Nakayama categories and Cohen-Macaulay cohomology Bibliography Index.